date()
## [1] "Sat Nov 26 09:14:30 2022"
library(MASS)
# load the data
data("Boston")
# check structure and dimensions
str(Boston)
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506 14
The dataset contains housing values in suburbs of Boston. It can be downloaded from R’s MASS package, and contains 506 observations of 14 variables. More information on the data and variables can be found here: https://stat.ethz.ch/R-manual/R-devel/library/MASS/html/Boston.html
library(tidyr)
# check summary of data
summary(Boston)
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08205 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
# draw pairs plot of the variables
pairs(Boston)
# calculate the correlation matrix and round it
cor_matrix <- cor(Boston)
cor_matrix %>% round(2)
## crim zn indus chas nox rm age dis rad tax ptratio
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58 0.29
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31 -0.39
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72 0.38
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04 -0.12
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67 0.19
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29 -0.36
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51 0.26
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53 -0.23
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91 0.46
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00 0.46
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46 1.00
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44 -0.18
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54 0.37
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47 -0.51
## black lstat medv
## crim -0.39 0.46 -0.39
## zn 0.18 -0.41 0.36
## indus -0.36 0.60 -0.48
## chas 0.05 -0.05 0.18
## nox -0.38 0.59 -0.43
## rm 0.13 -0.61 0.70
## age -0.27 0.60 -0.38
## dis 0.29 -0.50 0.25
## rad -0.44 0.49 -0.38
## tax -0.44 0.54 -0.47
## ptratio -0.18 0.37 -0.51
## black 1.00 -0.37 0.33
## lstat -0.37 1.00 -0.74
## medv 0.33 -0.74 1.00
# visualize the correlation matrix
library(corrplot)
## corrplot 0.92 loaded
corrplot(cor_matrix, method="circle")
The summary of the data shows means, medians and ranges of the different variables, for example median number of rooms per dwelling is 6.2 (range 3.6-8.8).
Tax rate and accessibility to radial highways seem to be strongly positively correlated. Median value and lower status of the population are negatively correlated. The plotted correlation matrix shows also other positive and negative correlations. The darker blue a sphere is, the stronger the positive correlation between variables, and the darker red a sphere is, the stronger the negative correlation.
library(dplyr)
##
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
##
## select
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)
## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865
# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
boston_scaled$crim <- as.numeric(boston_scaled$crim)
# create a quantile vector of crim
bins <- quantile(boston_scaled$crim)
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, label = c("low", "med_low", "med_high", "high"), include.lowest = TRUE)
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
The variables are now transformed on the same scale, so now we can compare them.
# linear discriminant analysis
lda.fit <- lda(crime ~., data = train)
# print the lda.fit object
lda.fit
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2326733 0.2549505 0.2500000 0.2623762
##
## Group means:
## zn indus chas nox rm age
## low 0.8939510 -0.9754278 -0.104792887 -0.8485044 0.43405146 -0.8736005
## med_low -0.1246578 -0.2867301 -0.004759149 -0.5745673 -0.10386054 -0.3349787
## med_high -0.3929954 0.2851589 0.234426408 0.4297635 0.09979558 0.4504816
## high -0.4872402 1.0170298 -0.086616792 1.0343800 -0.41806290 0.7834562
## dis rad tax ptratio black lstat
## low 0.8222879 -0.6825367 -0.7269244 -0.45365083 0.37315387 -0.750377064
## med_low 0.3216926 -0.5592799 -0.4881197 -0.09336749 0.32650788 -0.133788700
## med_high -0.4346438 -0.4360651 -0.2968440 -0.31971579 0.08686171 0.007848367
## high -0.8364898 1.6390172 1.5146914 0.78181164 -0.76110497 0.840072897
## medv
## low 0.52446690
## med_low 0.01469537
## med_high 0.20108198
## high -0.67738454
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.144675133 0.62018262 -0.96325232
## indus 0.150676733 -0.60868188 0.42780791
## chas -0.042350476 -0.02561201 0.04412303
## nox 0.372367898 -0.53473272 -1.44290314
## rm 0.008221839 -0.08137991 -0.09785932
## age 0.238992718 -0.30502986 -0.07271449
## dis -0.029520974 -0.16386419 0.18039555
## rad 3.478432985 0.78820916 0.10942887
## tax -0.091702733 0.18587134 0.35045796
## ptratio 0.125339980 0.05651590 -0.35584810
## black -0.096920528 0.03746421 0.13243900
## lstat 0.148232788 -0.15750481 0.32515880
## medv 0.053729282 -0.34615529 -0.27116387
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9511 0.0369 0.0120
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 2)
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
## predicted
## correct low med_low med_high high
## low 21 11 1 0
## med_low 3 16 4 0
## med_high 0 11 12 2
## high 0 0 0 21
It would seem that with high crime rate the predictions are most accurate. With low to medium and medium to high there is most inaccuracy.
library(ggplot2)
set.seed(123)
data(Boston)
boston_scaled2 <- as.data.frame(scale(Boston))
# euclidean distance matrix
dist_eu <- dist(boston_scaled2)
summary(dist_eu)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970
# k-means clustering
km <- kmeans(boston_scaled2, centers = 3)
# plot part of the Boston dataset with clusters
pairs(boston_scaled2[6:10], col = km$cluster)
# determine the maximum number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled2, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <- kmeans(boston_scaled2, centers = 2)
# plot the Boston dataset with clusters
pairs(boston_scaled2, col = km$cluster)
# plot part of the Boston dataset with clusters
pairs(boston_scaled2[6:10], col = km$cluster)
From plotting the total within sum of squares we see that around two the value changes quite a lot, so the appropriate number of cluster would be two. The data seems to divide nicely between two clusters according to most variables.
set.seed(123)
data(Boston)
boston_scaled3 <- as.data.frame(scale(Boston))
# k-means clustering
km2 <- kmeans(boston_scaled3, centers = 3)
# linear discriminant analysis
lda.fit2 <- lda(km2$cluster ~., data = boston_scaled3)
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(km2$cluster)
# plot the lda results
plot(lda.fit2, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit2, myscale = 4)
Variables age, black, and tax seem to be the most influential linear separators for the clusters.
library(plotly)
##
## Attaching package: 'plotly'
## The following object is masked from 'package:ggplot2':
##
## last_plot
## The following object is masked from 'package:MASS':
##
## select
## The following object is masked from 'package:stats':
##
## filter
## The following object is masked from 'package:graphics':
##
## layout
lda.fit3 <- lda(crime ~., data = train)
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)
## [1] 404 13
dim(lda.fit3$scaling)
## [1] 13 3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit3$scaling
matrix_product <- as.data.frame(matrix_product)
classes <- as.numeric(train$crime)
train2 <- dplyr::select(train, -crime)
km3 <- kmeans(train2, centers = 4)
clusters <- as.numeric(km3$cluster)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=classes)
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color=clusters)
The cluster that is on the right-hand side (at approximately x=-4 and y=2) looks quite same in both plots. In the first plot the clusters are more defined, in the second one they mix more with each other.